Calculus/Regaridng Integral

Let me put my understanding of Integral first. I thought Integral (I could not find the symbol in my computer) of dx means addition of minute dx to form x. I would like to know how could I understand Integral of x/2 dx. I know that the answer is x^3/3. But I wanted to know how addition of minute x/2 led to x^3/3.

Please explain.

ANSWER: It can't. If it appears, there is a misprint in the book.
In the following answers, I assume that we have the integral, the function given,
and a dx at the end. Also, we need a + C with no limits.

If the x/2 was really x^2, it was done correctly.

If the problem is to integrate x/2, the answer is x^2/4 + C.

If the answer is x^3/3 + C, the function integrated was x^2.

The answer to any integral of powers of x always raises the power of x by only 1 and then divides by x. For example,
the integral of x is x^2/2 + C,
the integral of x^2 is x^3/3 + C,
the integral of x^3 is x^4/4 + C,
etc.

---------- FOLLOW-UP ----------

QUESTION: Hi Scott,

Thanks for you reply. Yes it is a mistake in my part. ok.. integral of x is x^2/2 + C. I want to understand the meaning of this. If I add minute xdx, how did it led to x^2/2. For example lets say x=2, then I would think integral of xdx as addition of 2dx,2dx,...and so on.. but the answer could not be x^2/2. Please clarify how addition of minute 2dx led to x^2/2.

regards.

ANSWER: Sorry about the time, but I couldn't get on the PC lately.

That just the way calculus goes. It always adds one to x when integrating.
To find the are of a rectangle, the height is multiplied by the width, which is x.
To find the area beneath a line, the height is multiplied by infinitely small values of length.
That is where the dx and integration comes from.

Let's look at the integral of x^2 from 2 to 5.
If we divide the area into n rectangles, each of them would have a width of 3/n.

The left hand side of each of these rectangles would be sum(k=1 to n)[(3/n)[(k-1)/n)]^2].
The right hand side of each of these rectangles would be sum(k=1 to n)[(3/n)((k/n)^2).

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